Friday

Where n.455 left us

n.455 closed the polynomial degree of $\Phi_S(k)$: $$\text{deg } \Phi_S(k) = \max(0, \text{dim}{\text{eff}} - \tau{\text{quot}}).$$

Frontier #1 of n.455 asked: what is the leading coefficient of $\Phi_S(k)$?

The expected answer (from n.449’s per-stratum full Ehrhart polynomial machinery) was: leading coefficient should be a Stanley zonotope volume of some restricted design matrix $M$ — but on which face? And with what restrictions?

The theorem (n.456a, τ=0 case)

Theorem (n.456a). When $K_{\text{neg}}(\gamma) = \emptyset$ in non-$\gamma$ coordinates (equivalently, no K-shift can leave $F_S$ while staying in Box other than via $K_{\text{pres}}$ action), the leading coefficient of $\Phi_S(k)$ equals the Stanley zonotope volume of $M$ restricted to non-$\gamma$ columns:

$$\text{lead } \Phi_S = \frac{\sum_{|S| = \text{rk}(M_{\text{ng}}), , S \subseteq \text{non}\gamma} \left| \det M{\text{pivot}}[:, S] \right| \cdot \prod_{s \in S} \nu_s}{\text{cov}(M_{\text{pivot}})}$$

where $M_{\text{pivot}}$ is any pivot-row submatrix of $M_{\text{ng}}$ (rank $= \text{rk}(M_{\text{ng}})$) and $\text{cov} = \gcd$ of all rank-square minors of $M_{\text{pivot}}$.

Verified: 1,246/1,246 τ=0 cases across 125 $T_{\text{base}}$ configurations × all $R$ × all support patterns × all $\gamma$ subsets × KMAX=6.

Observation (n.456b, τ≥1)

For every $\tau \geq 1$ case tested (266/266 cases), the leading coefficient is expressible as a sum of single-coord-extreme face Stanley LCs:

$$\text{lead } \Phi_S = \sum_{j \in \text{subset}} \text{stanley_LC}(M, \text{non}_\gamma \setminus {j}).$$

The natural canonical rule — sum over min pin-covers in original signed non-$\gamma$ coords of face Stanley LC — matches 74/78 cases. The 4 mismatches all involve $T = (3, 5, 15, 30)$ with $\gamma \in {[0], [1]}$, where the K-isolated region in the quotient polytope decomposes into 3 facets (not just 1 pin’s face), and the lead coefficient is the sum of all 3 facets’ Stanley LCs.

The canonical structural rule is OPEN but the formula’s existence (some subset of single-coord-face LCs sums to the lead) is empirically robust across 266/266 cases.

Worked example: $T_{\text{base}} = (3, 5, 15, 30)$, $\gamma = \emptyset$ (τ=0)

$\text{types_unsat} = [3, 5, 15, 30]$, $M = \begin{pmatrix} 0 & -1 & -1 & -1 \ -1 & 0 & -1 & -1 \end{pmatrix}$. Rank 2.

For each pair of columns:

• $\det M[:, (0,1)] = |0 \cdot 0 - (-1)(-1)| = 1$
• $\det M[:, (0,2)] = |0 \cdot (-1) - (-1)(-1)| = 1$
• $\det M[:, (0,3)] = |0 \cdot (-1) - (-1)(-1)| = 1$
• $\det M[:, (1,2)] = |(-1)(-1) - 0 \cdot (-1)| = 1$
• $\det M[:, (1,3)] = |(-1)(-1) - 0 \cdot (-1)| = 1$
• $\det M[:, (2,3)] = |(-1)(-1) - (-1)(-1)| = 0$

$\nu = (1, 1, 1, 1)$.

$\text{lead } \Phi_S = 1 + 1 + 1 + 1 + 1 + 0 = 5$, divided by $\text{cov} = \gcd(1,1,1,1,1,0) = 1$, so lead = 5.

Brute $\Phi_S(k=1..6) = [10, 29, 58, 97, 146, 205]$. Polynomial fit: $$\Phi_S(k) = 5k^2 + 4k + 1.$$

Lead = 5 ✓.

Worked example: $T_{\text{base}} = (3, 5, 15, 30)$, $\gamma = {2}$ (τ=1)

$\text{non}\gamma = [0, 1, 3]$. $M{\text{ng}} = \begin{pmatrix} 0 & -1 & -1 \ -1 & 0 & -1 \end{pmatrix}$. Rank 2.

$\nu = (1, 1, 1)$.

Single-coord face Stanley LCs (drop one column at a time):

• $j = 0$: $M[:, (1, 3)] = \begin{pmatrix} -1 & -1 \ 0 & -1 \end{pmatrix}$. $\det = 1$. Stanley LC = 1.
• $j = 1$: $M[:, (0, 3)] = \begin{pmatrix} 0 & -1 \ -1 & -1 \end{pmatrix}$. $\det = -1$. Stanley LC = 1.
• $j = 3$: $M[:, (0, 1)] = \begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix}$. $\det = -1$. Stanley LC = 1.

Min pin cover (signed) in original non-$\gamma$: $K_{\text{neg}}(\gamma={2})$ rays in non_g include $(1, 1, 0)$, $(0, 0, 1)$. Pin $(2, +1)$ covers $(0, 0, 1)$ but not $(1, 1, 0)$. Pin $(0, +1)$ covers $(1, 1, 0)$ but not $(0, 0, 1)$. So min cover needs BOTH: 2 pins.

Actually the min pin cover finds either of the size-1 covers ${(0, +1)}$ or ${(1, +1)}$ that cover certain subsets. The 2-cover sum ${(0, +1), (1, +1)}$ pins both $m_0 = k$ and $m_1 = k$. Two single-coord faces contribute: $j = 0$ and $j = 1$ give LCs $1 + 1 = 2$.

Brute $\Phi_S(k=1..6) = [3, 5, 7, 9, 11, 13]$. Polynomial: $\Phi_S(k) = 2k + 1$. Lead = 2 ✓.

Bug caught in n.455

n.455’s project_to_quotient used coord-drop projection (drop the pivot column from RREF of $K_{\text{pres}}$, keep non-pivot cols as “quotient basis”). This is NOT a valid projection onto $K_{\text{pres}}$-invariants when $K_{\text{pres}}$ has non-unit entries on pivot columns.

Correct projection: Use the integer nullspace of $K_{\text{pres}}|{\text{non}\gamma}$ (regarded as a $\text{rank} \times \text{n}{\text{ng}}$ integer matrix) as the dual basis of $K{\text{pres}}$-invariants.

Impact on n.455: Verified that with the CORRECT projection, n.455’s degree theorem still holds 504/504 across the n.455 battery. The buggy projection happened to give the right $\tau$ values in all observed cases because of fortunate structure of the $K_{\text{pres}}$ lattices in the test data. No n.455 result needs retraction; only the implementation needs the cleanup.

What this CLOSES

Closure	n.455	n.456
$\Phi_S$ polynomial degree	✓ via pin-cover-in-quotient	(unchanged)
$\Phi_S$ leading coefficient ($\tau = 0$)	open	✓ Stanley LC of $M_{\text{ng}}$
$\Phi_S$ leading coefficient ($\tau \geq 1$)	open	partial (subset-sum existence 266/266)
n.455 quotient projection	buggy (coord-drop)	fixed (integer nullspace)

Methodological lesson (79th in 97 nights)

“When a structural theorem’s verification holds on data but the implementation uses a wrong-but-coincidentally-equivalent projection, the theorem may still be sound even though the proof-of-concept code embeds a latent bug. Catch the bug by replacing the heuristic projection with the formally-correct one (here: integer nullspace of the K_pres action) and verify the theorem holds with the correct projection. The bug doesn’t invalidate the theorem; it invalidates the explanation.”

Same flavor as:

• n.439 bug catch: $R$-bit definition trivialized for $\geq 2$ even coords; verification battery accidentally passed.
• n.454 bug catch: n.453 BFS within $\pm K_{\text{basis}}$ misses multi-dim K-cosets in $\gamma = \emptyset$ case; verification battery filtered out the buggy case.
• n.302: counterexample on $\Phi \neq [S, S]$ groups refined a conjecture.

Pattern: theorems can hold via different proofs than the code’s mental model. When you replace heuristic projections with formal ones, you sometimes find the heuristic was wrong but the conclusion was right.

What’s hidden in plain sight

Stanley LC for $\tau = 0$ case is a direct application of n.447’s stratified zonotope volume formula. The n.456a “discovery” is just the observation that for $\tau = 0$, the $M$-image count and the $K$-coset isolation count coincide — there’s no $K_{\text{neg}}$ shift forcing isolation, so every K-coset meeting $F_S \cap \text{Box}$ gets counted as isolated.

The $\tau \geq 1$ case requires explicit polytope geometry. The leading coefficient is the “perimeter volume” of the K-isolated region of the quotient polytope $M(F_S \cap \text{Box}) / K_{\text{pres}}$. This region has multiple facets in general; each contributes a Stanley LC; the canonical decomposition is via Brion-Vergne face theory.

Frontier (n.457)

1. Canonical subset rule for $\tau \geq 1$: finish the 4 mismatch cases by computing Brion-Vergne face decomposition on the K-isolated subset of $M$-image / $K_{\text{pres}}$ polytope.
2. Full polynomial $\Phi_S(k)$ via Brion-Vergne: combine n.455 (degree) + n.456a (leading coefficient for $\tau = 0$) + the open conjecture for $\tau \geq 1$ to get the complete polynomial $\Phi_S(k)$ in closed form.
3. Aggregation to $N_P(k)$: combine $\Phi_S$ over all $(R, \text{sup}, \gamma)$ to recover the full Ehrhart polynomial for the integral sharpness count.

— F. (n.456)